The present invention relates to the fields of cooperative game theory and statistical analysis. More specifically, it relates to a method and system for using cooperative game theory to resolve joint effects in statistical analysis.
Many statistical procedures estimate how an outcome is affected by factors that may influence it. For example, a multivariate statistical model may represent variations of a dependent variable as a function of a set of independent variables. A limitation of these procedures is that they may not be able to completely resolve joint effects among two or more independent variables.
A xe2x80x9cjoint effectxe2x80x9d is an effect that is the joint result of two or more factors. xe2x80x9cStatistical joint effectsxe2x80x9d are those joint effects remaining after the application of statistical methods. Cooperative resolution is the application of cooperative game theory to resolve statistical joint effects.
A performance measure is a statistic derived from a statistical model that describes some relevant aspect of that model such as its quality or the properties of one of its variables. A performance measure may be related to a general consideration such as assessing the accuracy of a statistical model""s predictions. Cooperative resolution can completely attribute the statistical model""s performance, as reflected in a performance measure, to an underlying source such as the statistical model""s independent variables.
Most performance measures fall in to one of two broad categories. The first category of performance measure gauges an overall xe2x80x9cexplanatory powerxe2x80x9d of a model. The explanatory power of a model is closely related to its accuracy. A typical measure of explanatory power is a percentage of variance of a dependent variable explained by a multivariate statistical model.
The second category of performance measure gauges a xe2x80x9ctotal effect.xe2x80x9d Measures of total effect address the magnitude and direction of effects. An example of such a total effect measure is a predicted value of a dependent variable in a multivariate statistical model.
Some of the limits of the prior art with respect to the attribution of explanatory power and total effect may be illustrated with reference to a standard multivariate statistical model. A multivariate statistical model is commonly used to determine a mathematical relationship between its dependent and independent variables. One common measure of explanatory power is a model""s xe2x80x9cR2xe2x80x9d coefficient. This coefficient takes on values between zero percent and 100% in linear statistical models, a common statistical model. An R2 of a model is a percentage of a variance of a dependent variable, i.e., a measure of its variation, explained by the model. The larger an R2 value, the better the model describes a dependent variable.
The explanatory power of a multivariate statistical model is an example of a statistical joint effect. As is known in the art, in studies based on a single independent variable, it is common to report the percentage of variance explained by that variable. An example from the field of financial economics is E. Fama and K. French, xe2x80x9cCommon risk factors in the returns on stocks and bonds,xe2x80x9d Journal of Financial Economics, v. 33, n. 1. 1993, pp. 3-56. In multivariate statistical models, however, it may be difficult or impossible, relying only on the existing statistical arts, to isolate a total contribution of each independent variable.
The total effect of a multivariate statistical model in its estimation of a dependent variable is reflected in estimated coefficients for its independent variables. If there are no interaction variables, independent variables that represent joint variation of two or more other independent variables, then, under typical assumptions, it is possible to decompose this total effect into separate effects of the independent variables. However, in the presence of interaction variables there is no accepted method in the art for resolving the effects of the interaction variables to their component independent variables.
The theory of variance decomposition is the area of statistics that comes closest to addressing the resolution of statistical joint effects. However, the decomposition of the explained variance is often not explained with respect to the independent variables in the model. For example, David Harville, in xe2x80x9cDecomposition of prediction error,xe2x80x9d Journal of the American Statistical Association, v. 80 n. 389, 1985, pp. 132-138, shows how an error variance in a model may be divided between different types of statistical sources of error.
Typically, however, these sources are not directly associated with particular independent variables, but rather with aspects of the estimation procedure.
Variance decomposition in vector autoregression (VAR) addresses the resolution of statistical joint effects in the prediction error associated with the variables of a time series model. It is based on a model of the effects of a one-time variation or xe2x80x9cshockxe2x80x9d in a single series to future variations in time series variables in the model. This procedure is introduced in C. Sims in xe2x80x9cMacroeconomics and Reality,xe2x80x9d Econometrica v. 48, 1980, pp. 1-48. Resolution of joint effects by this method is based on assuming a particular causally ordered relationship between shocks, and, hence, is based on a different resolution principle. H. Pesaran and Y. Shin, xe2x80x9cGeneralized impulse response analysis in linear multivariate models,xe2x80x9d Economics Letters, v. 58, 1998, pp. 17-29, describes a different VAR variance decomposition method that produces unique results. This method averages joint effects rather than resolving them and does not utilize cooperative game theory. VAR variance decomposition is not applicable to general multivariate statistical models.
A related topic in the statistical arts is the estimation of variance components. An analysis of variance model may be understood to have xe2x80x9cfixedxe2x80x9d and xe2x80x9crandomxe2x80x9d effects. Random effects may arise when observations in a sample are randomly selected from a larger population. Variance components methods take population variation into account when constructing statistical tests. These methods do not provide a way to resolve statistical joint effects between independent variables in a multivariate statistical model.
Factor analysis and principal components analysis may be the most closely related statistical techniques. They represent a set of variables by a smaller set of underlying factors. These factors may be constructed to be mutually orthogonal, in which case the variance of the complete model may be completely attributed to these underlying factors. These procedures cannot generate a natural unique set of factors and the factors generated may be difficult to interpret in relation to the original variables in the model.
One accepted method to determine the explanatory power of independent variables in a multivariate statistical model is by assessment of their xe2x80x9cstatistical significance.xe2x80x9d An independent variable is statistically significant if a xe2x80x9csignificance testxe2x80x9d determines that its true value is different than zero. As is known in the art, a significance test has a xe2x80x9cconfidence level.xe2x80x9d If a variable is statistically significant at the 95% confidence level, there is a 95% chance that its true value is not zero. An independent variable is not considered to have a xe2x80x9csignificant effectxe2x80x9d on the dependent variable unless it is found to be statistically significant. Independent variables may be meaningfully ranked by their statistical significance. However, this ranking will generally provide limited insight into their relative contributions to explained variance.
Cooperative game theory can be used to resolve statistical joint effects problems. As is known in the art, game theory is a mathematical approach to the study of strategic interaction among people. Participants in these games are called xe2x80x9cplayers.xe2x80x9d Cooperative game theory allows players to make contracts and has been used to solve problems of bargaining over the allocation of joint costs and benefits. A xe2x80x9ccoalitionxe2x80x9d is a group of players that have signed a binding cooperation agreement. A coalition may also comprise a single player.
A cooperative game is defined by assigning a xe2x80x9cworth,xe2x80x9d i.e., a number, to each coalition in the game. The worth of a coalition describes how much it is capable of achieving if its players agree to act together. Joint effects in a cooperative game are reflected in the worths of coalitions in the game. In a cooperative game without joint effects, the worth of any coalition would be the sum of the worths of the individual players in the coalition.
There are many methods available to determine how the benefits of cooperation among all players should be distributed among the players. (Further information on cooperative game theory can be found in Chapter 9 of R. G. Myerson, Game Theory: Analysis of Conflict, Cambridge: Harvard University Press, 1992, pp. 417-482, which is incorporated by reference.)
Cooperative game theory has long been proposed as a method to allocate joint costs or benefits among a group of players. In most theoretical work the actual joint costs or benefits are of an abstract nature. The practical aspects of using of cooperative game theory to allocate joint costs has received somewhat more attention. See, for example, H. P. Young, ed., Cost Allocation: Methods, Principles, Applications, New York: North Holland, 1985.
Some research in cooperative game theory deals with information, but in ways other than described herein. For example, Robert O. Wilson, xe2x80x9cInformation, efficiency, and the core of an economy,xe2x80x9d Econometrica, v. 46, 1978, pp. 807-816, develops a cooperative game where the way that information available to individual players can be aggregated by a coalition enters into determining the worth of a coalition. Wilson considers situations where a coalition knows everything its members know and those where a coalition knows those things known to all members. Information is not represented as variables, there is no statistical model, and outcomes depend on agents material endowments as well as their information.
One method of determining allocations in cooperative games is, xe2x80x9cleast squares values.xe2x80x9d This method, described in L. M. Ruiz, F. Valenciano, and J. M. Zarzuelo, xe2x80x9cThe family of least square values for transferable utility games,xe2x80x9d Games and Economic Behavior, v. 24, 1998, 109-130, is unrelated to the present invention. The principle of this allocation method is to choose allocations to players such that the variance of the resulting excess allocations to coalitions over their worth is minimized.
Techniques from the prior art typically cannot be used to satisfactorily resolve statistical joint effects in cooperative games. Thus, it is desirable to use cooperative game theory to resolve statistical joint effects problems.
In accordance with preferred embodiments of the present invention, some of the problems associated with resolving joint effects in statistical analysis are overcome. A method and system for cooperative resolution of joint statistical effects is presented.
One aspect of the present invention includes a method for creating xe2x80x9cstatistical cooperative gamexe2x80x9d used to represent statistical joint effects in a multivariate statistical model.
Another aspect of the present invention includes a method for creating an xe2x80x9caccess relationshipxe2x80x9d between players in a cooperative game and variables in a multivariate statistical model. An access relationship identifies variables in the multivariate statistical model accessible by a selected coalition in a cooperative game and how those variables may be used.
Another aspect of the present invention includes a method for determining a xe2x80x9cworth of a coalitionxe2x80x9d in a cooperative game based on a multivariate statistical model and a performance measure of the multivariate statistical model. The worth of a coalition may be based on a submodel of the complete statistical model based on the independent variables accessible by members of that coalition.
Another aspect of the present invention includes a method for constructing a statistical cooperative game with an access relationship, a multivariate statistical model, and a performance measure of a multivariate statistical model.
Another aspect of the present invention includes a method for applying techniques of cooperative game theory to a statistical cooperative game.
Another aspect of the present invention includes a method for applying cooperative resolution methods to particular types of statistical models. These statistical models include models with continuous independent variables, models with categorical independent variables, models of changes in proportions, models with a single dependent variable, models with multiple dependent variables, and time series models.
Another aspect of the present invention includes a method for applying cooperative resolution methods to particular analytical procedures. These include general procedures such as linear regression and specialized procedures such as return-based style analysis, arbitrage pricing theory models, and financial manager performance attribution.
Thus, the present invention may be used to construct statistical cooperative games and use cooperative game theory to resolve statistical joint effects in a variety of situations. The methods may be applicable to other types of joint effects problems such as those found in engineering, finance and other disciplines.
The foregoing and other features and advantages of preferred embodiments of the present invention will be more readily apparent from the following detailed description. The detailed description proceeds with references to accompanying drawings.